The "arrowed" problems should be noted for later work.
Exercise (PageIndex{1})
Show that (E^{2}) becomes a metric space if distances (
ho(overline{x}, overline{y})) are defined by
(a) (
ho(overline{x}, overline{y})=leftx_{1}y_{1}
ight+leftx_{2}y_{2}
ight) or
(b) (
ho(overline{x}, overline{y})=max left{leftx_{1}y_{1}
ight,leftx_{2}y_{2}
ight
ight}),
where (overline{x}=left(x_{1}, x_{2}
ight)) and (overline{y}=left(y_{1}, y_{2}
ight) .) In each case, describe (G_{overline{0}}(1)) and (S_{overline{0}}(1) .) Do the same for the subspace of points with nonnegative coordinates.
Exercise (PageIndex{2})
Prove the assertions made in the text about globes in a discrete space. Find an empty sphere in such a space. Can a sphere contain the entire space?
Exercise (PageIndex{3})
Show that ( ho) in Examples ((3)) and ((5)) obeys the metric axioms.
Exercise (PageIndex{4})
Let (M) be the set of all positive integers together with the "point" (infty .) Metrize (M) by setting
[
ho(m, n)=leftfrac{1}{m}frac{1}{n}
ight, ext { with the convention that } frac{1}{infty}=0.
]
Verify the metric axioms. Describe (G_{infty}left(frac{1}{2}
ight), S_{infty}left(frac{1}{2}
ight),) and (G_{1}(1)).
Exercise (PageIndex{5})
(Rightarrow 5 .) Metrize the extended real number system (E^{*}) by
[
ho^{prime}(x, y)=f(x)f(y),
]
where the function
[
f : E^{*} underset{ ext { onto }}{longrightarrow}[1,1]
]
is defined by
[
f(x)=frac{x}{1+x} ext { if } x ext { is finite, } f(infty)=1, ext { and } f(+infty)=1.
]
Compute (
ho^{prime}(0,+infty),
ho^{prime}(0,infty),
ho^{prime}(infty,+infty),
ho^{prime}(0,1),
ho^{prime}(1,2),) and (
ho^{prime}(n,+infty) .) Describe (G_{0}(1), G_{+infty}(1),) and (G_{infty}left(frac{1}{2}
ight) .) Verify the metric axioms (also when infinities are involved).
Exercise (PageIndex{6})
(Rightarrow 6 .) In Problem (5,) show that the function (f) is one to one, onto ([1,1],) and increasing; i.e.
[
x
Also show that the (f) image of an interval ((a, b) subseteq E^{*}) is the interval ((f(a), f(b)) .) Hence deduce that globes in (E^{*}) (with (
ho^{prime}) as in Problem 5) are intervals in (E^{*}) (possibly infinite).
[Hint: For a finite (x,) put
[
y=f(x)=frac{x}{1+x}.
]
Solving for (x) (separately in the cases (x geq 0) and (x<0 ),) show that
[
(forall y in(1,1)) quad x=f^{1}(y)=frac{y}{1y};
]
thus (x) is uniquely determined by (y,) i.e., (f) is one to one and ontoeach (y in(1,1)) corresponds to some (x in E^{1} .) (How about (pm 1 ? ))
To show that (f) is increasing, consider separately the three cases (x<0
Exercise (PageIndex{7})
Continuing Problems 5 and (6,) consider (left(E^{1}, ho^{prime} ight)) as a subspace of (left(E^{*}, ho^{prime} ight)) with ( ho^{prime}) as in Problem (5 .) Show that globes in (left(E^{1}, ho^{prime} ight)) are exactly all open intervals in (E^{*} .) For example, ((0,1)) is a globe. What are its center and radius under ( ho^{prime}) and under the standard metric ( ho ?)
Exercise (PageIndex{8})
Metrize the closed interval ([0,+infty]) in (E^{*}) by setting
[
ho(x, y)=leftfrac{1}{1+x}frac{1}{1+y}
ight ,
]
with the conventions (1+(+infty)=+infty) and (1 /(+infty)=0 .) Verify the metric axioms. Describe (G_{p}(1)) for arbitrary (p geq 0).
Exercise (PageIndex{9})
Prove that if (
ho) is a metric for (S,) then another metric (
ho^{prime}) for (S) is given by
(i) (
ho^{prime}(x, y)=min {1,
ho(x, y)});
(ii) (
ho^{prime}(x, y)=frac{
ho(x, y)}{1+
ho(x, y)}).
In case ((mathrm{i}),) show that globes (G_{p}(varepsilon)) of radius (varepsilon leq 1) are the same under (
ho) and (
ho^{prime} .) In case (ii), prove that any (G_{p}(varepsilon)) in ((S,
ho)) is also a globe (G_{p}left(varepsilon^{prime}
ight)) in (left(S,
ho^{prime}
ight)) of radius
[
varepsilon^{prime}=frac{varepsilon}{1+varepsilon},
]
and any globe of radius (varepsilon^{prime}<1) in (left(S,
ho^{prime}
ight)) is also a globe in ((S,
ho) .) (Find the converse formula for (varepsilon) as well!)
[Hint for the triangle inequality in (ii): Let (a=
ho(x, z), b=
ho(x, y),) and (c=
ho(y, z)) so that (a leq b+c .) The required inequality is
[
frac{a}{1+a} leq frac{b}{1+b}+frac{c}{1+c}.
]
Simplify it and show that it follows from (a leq b+c . ])
Exercise (PageIndex{10})
Prove that if (left(X,
ho^{prime}
ight)) and (left(Y,
ho^{prime prime}
ight)) are metric spaces, then a metric (
ho) for the set (X imes Y) is obtained by setting, for (x_{1}, x_{2} in X) and (y_{1}, y_{2} in Y),
(i) (
holeft(left(x_{1}, y_{1}
ight),left(x_{2}, y_{2}
ight)
ight)=max left{
ho^{prime}left(x_{1}, x_{2}
ight),
ho^{prime prime}left(y_{1}, y_{2}
ight)
ight} ;) or
(ii) (
holeft(left(x_{1}, y_{1}
ight),left(x_{2}, y_{2}
ight)
ight)=sqrt{
ho^{prime}left(x_{1}, x_{2}
ight)^{2}+
ho^{prime prime}left(y_{1}, y_{2}
ight)^{2}}).
[Hint: For brevity, put (
ho_{12}^{prime}=
ho^{prime}left(x_{1}, x_{2}
ight),
ho_{12}^{prime prime}=
ho^{prime prime}left(y_{1}, y_{2}
ight),) etc. The triangle inequality in (ii),
[
sqrt{left(
ho_{13}^{prime}
ight)^{2}+left(
ho_{13}^{prime prime}
ight)^{2}} leq sqrt{left(
ho_{12}^{prime}
ight)^{2}+left(
ho_{12}^{prime prime}
ight)^{2}}+sqrt{left(
ho_{23}^{prime}
ight)^{2}+left(
ho_{23}^{prime prime}
ight)^{2}},
]
is verified by squaring both sides, isolating the remaining square root on the right side, simplifying, and squaring again. Simplify by using the triangle inequalities valid in (X) and (Y,) i.e.,
[
ho_{13}^{prime} leq
ho_{12}^{prime}+
ho_{23}^{prime} ext { and }
ho_{13}^{prime prime} leq
ho_{12}^{prime prime}+
ho_{23}^{prime prime}.
]
Reverse all steps, so that the required inequality becomes the last step. (])
Exercise (PageIndex{11})
Prove that
[

ho(y, z)
ho(x, z) leq
ho(x, y)
]
in any metric space ((S,
ho) .)
[Caution: The formula (
ho(x, y)=xy,) valid in (E^{n},) cannot be used in ((S,
ho) .) Why? (])
Exercise (PageIndex{12})
Prove that
[
holeft(p_{1}, p_{2}
ight)+
holeft(p_{2}, p_{3}
ight)+cdots+
holeft(p_{n1}, p_{n}
ight) geq
holeft(p_{1}, p_{n}
ight).
]
[Hint: Use induction. (])